LEC ANA2 15

We’ve seen an example: $E\subseteq F\subseteq \overline{E}$, $E$ is path connected, $F$ is connected but not path connected, $\overline{E}$ is path connected.

[!Example]
Let $E=\{(x,\sin (1/x)):x\in (0,1]\}$. $\overline{E}=E\cup \{0\}\times [-1,1]$. $E$ is path connected. $E$, $\overline{E}$ are connected.

Claim: $\overline{E}$ is not path connected. Suppose there is a path $\gamma$ connecting $(0,0)$ and $(1,\sin (1))$. Let $K={\gamma }^{-1}(\{0\}\times [-1,1])\subseteq [0,1]$. $K$ is compact. Let $t_0=\sup K$. $t_0\in K$. For all $t>t_0$, $\gamma (t)\in E$. Thus, $P_x(\gamma (t_0))=0$, whereas $P_x(\gamma (t_0+1/k))>0$ for all $k>0$.

For $k>0$. there exists $n_k$ such that $t_0<\frac{1}{2\pi n_k+\theta }<t_0+1/k$. By the intermediate value theorem, there exists $t_k$ such that $P_x(\gamma (t_k))=\frac{1}{2\pi n_k+\theta }$. Let $t_k\to t_0$. $P_y(\gamma (t_k))\to \sin \theta$. Take $\theta$ such that $\sin \theta \ne P_y(\gamma (t_0))$.

[!Example]
We’ve seen that $G{L}_n(\mathbb{R})$ is not connected.

Claim: $G{L}_n(\mathbb{C})$ is connected.

[!Lemma]
Let $p$ be a complex polynomial in $n$ complex variables. Let $Z(p)$ be the zero set of $p$. ${\mathbb{C}}^n\setminus Z(p)$ is path connected.

[!proof]-

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Cantor set

to show $C$ is uncountable, show it is perfect.